Consider a sequence of independent and identically distributed random variables along with a specified set of k-vectors. We present an expression for E [T], the mean time until the last k observed random variables fall within this set. Not only can this expression often be used to obtain bounds on E[T], it also gives rise to an efficient way of approximating E[T] by a simulation. Specific lower and upper bounds for E[T] are also derived. These latter bounds are given in terms of a parameter, and a Markov chain Monte Carlo approach to approximate this parameter by a simulation is indicated. The results of this paper are illustrated by considering the problem of determining the mean time until a sequence of k-valued random variables has a run of size k that encompasses each value.

We consider a process that starts at height y, stays there for a time X0 ∼ exp(y) when it drops to a level Z1 ∼ U(0, y). Thereafter it stays at level Zn for time Xn ∼ exp(Zn), then drops to a level Zn+1 ∼ U(0,Zn). We investigate properties of this process, as well as the Poisson hyperbolic process which is obtained by randomizing the starting point y of the above process. This process is associated with a rate 1 Poisson process in the positive quadrant: Its path is the minimal RCLL decreasing step function through Poisson points in the positive quadrant. The finite dimensional distributions are then multivariate exponential in sense of Marshall-Olkin.

Consider a family of distributions with survival distributions that are log concave and stochastically increasing in a parameter over which it will be mixed. It is shown that a necessary and sufficient condition for a mixture over any such family to have an increasing failure rate is that the mixing distribution have an increasing failure rate. Some observations are given on generalizing this result as well as weakening the conditions on Prekopa's Theorem.

Since callers encountering busy signals often want to redial, modern communication networks have been designed to provide automatic redialing. Redialing services commonly have two parameters: a maximum number n of retries and a total duration τ over which retries are to be made. Typically, retries are made at evenly spaced time intervals of length τ/n until either the call succeeds or n retries have failed. This rule has an obvious intuitive appeal; indeed, among the main results of this paper are proofs that τ/n-spacing is optimal in certain basic models of called-number behavior. However, it is easy to find situations where τ/n-spacing is not optimal, as the paper verifies.

All of our models assume Poisson arrivals, but different assumptions are studied for the call durations; for a given mean, these are allowed to have the relatively high-variance exponential distribution or the zero-variance distribution concentrated at a point. We approximate the probability of success for the Erlang loss model with c > 1 trunks, and we calculate exact probabilities of success for the c = 1 Erlang model and the model with one trunk and constant call durations. For the latter model, we present two intriguing conjectures, one about the optimal choice of τ when n = 1 and one about the optimality of the τ/n-spacing policy. In spite of their apparent simplicity, these conjectures seem difficult to resolve. Finally, we study policies that continue redialing until they succeed, balancing a short mean wait against a small mean number of retries to success.

Control limit type policies are widely discussed in the literature, particularly regarding the maintenance of deteriorating systems. Previous studies deal mainly with stationary deterioration processes, where costs and transition probabilities depend only on the state of the system, regardless of its cumulative age. In this paper, we consider a nonstationary deterioration process, in which operation and maintenance costs, as well as transition probabilities “deteriorate” with both the system's state and its cumulative age. We discuss conditions under which control limit policies are optimal for such processes and compare them with those used in the analysis of stationary models.

Two maintenance models are examined: in the first (as in the majority of classic studies), the only maintenance action allowed is the replacement of the system by a new one. In this case, we show that the nonstationary results are direct generalizations of their counterparts in stationary models. We propose an efficient algorithm for finding the optimal policy, utilizing its control limit form. In the second model we also allow for repairs to better states (without changing the age). In this case, the optimal policy is shown to have the form of a 3-way control limit rule. However, conditions analogous to those used in the stationary problem do not suffice, so additional, more restrictive ones are suggested and discussed.

Formulas are derived for moments of the number of refused customers in a busy period for the M/GI/1/n and the GI/M/1/n queueing systems. As an interesting special case for the M/GI/1/n system, we note that the mean number is 1 when the mean interarrival time equals the mean service time. This provides a more direct argument for a result given in Abramov [1].

The move-to-front rule is applied on a model where several records are accessed each time. The records will then be placed in the front positions randomly or with the former relative order between themselves preserved. Equilibrium distributions are explored. Comparison of expected stationary search costs is carried out.

We give a new bound on the spectral gap of an ergodic time reversible Markov chain in term of the conductance of the chain.

We consider a K-node queueing system sharing a setup server. Each node has a node server, a waiting position, and a service position, and it behaves as an M/G/1/2 type queue except that each job in the waiting position requires a setup by the setup server to move to the service position. The service discipline of the setup server is nonpreemptive work-conserving random selection. The setup times have a common exponential distribution. The main purpose of this paper is to derive the stationary distribution for the K-node system. For each node, we construct a corresponding setup server queue (CSQ). The stationary distribution of the K-node system is given by a product form of the stationary distributions of CSQs. This result enables us to obtain the stationary distribution of a K-node system by analyzing individual CSQs.

Discrete-time queues are frequently used in telecommunication networks. This paper proposes a new method which is based only on probabilistic arguments. It derives the distributions of numbers of customers in the discrete-time systems Geom/G/1 and Geom/G/1/M. Though the derivation does not involve use of the transforms, the transforms may be obtained, if desired. Another advantage of this is that the numerical results obtained are stable as there are no negative signs involved in summations. Further, the method can be easily used to solve more complex problems in discrete- and continuous-time queues.