This article investigates the accuracy of approximations for the distribution of ordered m-spacings for i.i.d. uniform observations in the interval (0, 1). Several Poisson approximations and a compound Poisson approximation are studied. The result of a simulation study is included to assess the accuracy of these approximations. A numerical procedure for evaluating the moments of the ordered m-spacings is developed and evaluated for the most accurate approximation.

Let A1, A2, …, An and B1, B2,. ., BN be two sequences of events on the same probability space. Let mn(A) and mN(B), respectively, be the number of those Aj and Bj which occur. Let Si,j denote the joint ith binomial moment of mn(A) and jth binomial moment of mN(B), 0 ≤ i ≤ n, 0 ≤ j ≤ N. For fixed non-negative integers a and b, we establish both lower and upper bounds on the distribution P(mn(A) = r, mN(B) = u) by linear combinations of Si,j, 0 ≤ i ≤ a, 0 ≤ j ≤ b. When both a and b are even, all mentioned S¡,j are utilized in both the upper and the lower bound. In a set of remarks the results are analyzed and their relation to the existing literature, including the univariate case, is discussed.

Lagrangian distributions are reviewed from the viewpoint of the Galton-Watson process. They are related to the busy period in queuing systems and to the first visit in random walks.

A property of the distributions is remarked for the application to vacant vehicles in a new transit system. Combinatorial identities of multinomial and binomial coefficients and related recurrences are shown by a probabilistic method. Based on the identities and recurrences, random forests generated by the Poisson and geometric Galton-Watson processes are characterized.

Stochastic majorization is a tool that has been used in many areas of probability and statistics (such as multivariate statistical analysis, queueing theory and reliability theory) in order to obtain useful bounds and inequalities. In this paper we study the relations among several notions of stochastic majorization and stochastic convexity and obtain sufficient (and sometimes necessary) conditions which imply some of these notions. Extensions and generalizations of several results in the literature are obtained. Some examples and applications regarding stochastic comparisons of order statistics are also presented in order to illustrate the results of the paper.

Shock models based on Poisson processes have been used to derive univariate and multivariate exponential distributions. But in many applications, Poisson processes are not realistic models of physical shock processes because they have independent increments; expanded models that allow for possibly dependent increments are of interest. In this paper, univariate and bivariate Pólya urn schemes are used to derive models of shock sources. The life distributions obtained from these models form a large parametric family that includes the exponential distribution. Even in the univariate case these life distributions have not been widely used, though they form a large and flexible family. In the bivariate case, the family includes the bivariate exponential distributions of Marshall and Olkin as a special case.

The first-order autoregressive semi-Mittag-Leffler (SMLAR(1)) process is introduced and its properties are studied. As an illustration, we discuss the special case of the first-order autoregressive Mittag-Leffler (MLAR(1)) process.

Let {Fn}n ≧ 0 be a sequence of c.d.f. and let {Rn}n ≧ 1 be the sequence of record values in a non-stationary record model where after the (n − 1)th record the population is distributed according to Fn. Then the equidistribution of the nth population and the record increment Rn – Rn– 1 (i.e. Rn – Rn– 1~ Fn) characterizes Fn to have an exponentially decreasing hazard function. To be more precise Fn is the exponential distribution if the support of Rn– 1 generates a dense subgroup in and otherwise the entity of all possible solutions can be obtained in the following way: let for simplicity the above additive subgroup be any c.d.f. F satisfying F(0) = 0, F(1) < 1 can be chosen arbitrarily. Setting λ = – log(1 – F(1)), Fn(x) = 1 – F(x – [x])exp(–λ [x]) is an admissible solution coinciding with F on the interval [0, 1] ([x] denotes the integer part of x). Simple additional assumptions ensuring that Fn is either exponential or geometric are given. Similar results for exponential or geometric tail distributions based on the independence of Rn– 1 and Rn – Rn– 1 are proved.

The paper investigates stochastic processes directed by a randomized time process. A new family of directing processes called Hougaard processes is introduced. Monotonicity properties preserved under subordination, and dependence among processes directed by a common randomized time are studied. Results for processes subordinated to Poisson and stable processes are presented. Potential applications to shock models and threshold models are also discussed. Only Markov processes are considered.

In this paper we introduce the concept of repair replacement. Repair replacement is a maintenance policy in which items are preventively maintained when a certain time has elapsed since their last repair. This differs from age replacement where a certain amount of time has elapsed since the last replacement. If the last repair was a complete repair, repair replacement is essentially the same as age replacement. It is in the case of minimal repair that these two policies differ. We make comparison between various types of policies in order to determine when and under which condition one type of policy is better than another.

Likelihood ratios are used in computer simulation to estimate expectations with respect to one law from simulation of another. This importance sampling technique can be implemented with either the likelihood ratio at the end of the simulated time horizon or with a sequence of likelihood ratios at intermediate times. Since a likelihood ratio process is a martingale, the intermediate values are conditional expectations of the final value and their use introduces no bias.

We provide conditions under which using conditional expectations in this way brings guaranteed variance reduction. We use stochastic orderings to get positive dependence between a process and its likelihood ratio, from which variance reduction follows. Our analysis supports the following rough statement: for increasing functionals of associated processes with monotone likelihood ratio, conditioning helps. Examples are drawn from recursively defined processes, Markov chains in discrete and continuous time, and processes with Poisson input.

Optimization problems in cancer radiation therapy are considered, with the efficiency functional defined as the difference between expected survival probabilities for normal and neoplastic tissues. Precise upper bounds of the efficiency functional over natural classes of cellular response functions are found. The ‘Lipschitz' upper bound gives rise to a new family of probability metrics. In the framework of the ‘m hit-one target' model of irradiated cell survival the problem of optimal fractionation of the given total dose into n fractions is treated. For m = 1, n arbitrary, and n = 1, 2, m arbitrary, complete solution is obtained. In other cases an approximation procedure is constructed. Stability of extremal values and upper bounds of the efficiency functional with respect to perturbation of radiosensitivity distributions for normal and tumor tissues is demonstrated.

One expects, intuitively, that the total damage caused by an epidemic increases, in a certain sense, with the infection intensity exerted by the infectives during their lifelength. The original object of the present work is to make precise in which probabilistic terms such a statement does indeed hold true, when the spread of the disease is described by a collective Reed–Frost model and the global cost is represented by the final size and severity. Surprisingly, this problem leads us to introduce an order relation for -valued random variables, unusual in the literature, based on the descending factorial moments. Further applications of the ordering occur when comparing certain sampling procedures through the number of un-sampled individuals. In particular, it is used to reinforce slightly comparison results obtained earlier for two such samplings.

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:

(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.

(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.

We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.

Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

We consider the relationships among the stochastic ordering of random variables, of their random partial sums, and of the number of events of a point process in random intervals. Two types of result are obtained. Firstly, conditions are given under which a stochastic ordering between sequences of random variables is inherited by (vectors of) random partial sums of these variables. These results extend and generalize theorems known in the literature. Secondly, for the strong, (increasing) convex and (increasing) concave stochastic orderings, conditions are provided under which the numbers of events of a given point process in two ordered random intervals are also ordered.

These results are applied to some comparison problems in queueing systems. It is shown that if the service times in two M/GI/1 systems are compared in the sense of the strong stochastic ordering, or the (increasing) convex or (increasing) concave ordering, then the busy periods are compared for the same ordering. Stochastic bounds in the sense of increasing convex ordering on waiting times and on response times are provided for queues with bulk arrivals. The cyclic and Bernoulli policies for customer allocation to parallel queues are compared in the transient regime using the increasing convex ordering. Comparisons for the five above orderings are established for the cycle times in polling systems.

In a discrete-time renewal process {Nk, k = 0, 1, ·· ·}, let Zk and Ak be the forward recurrence time and the renewal age, respectively, at time k. In this paper, we prove that if the inter-renewal time distribution is discrete DFR (decreasing failure rate) then both {Ak, k = 0, 1, ·· ·} and {Zk, k = 0, 1, ·· ·} are monotonically non-decreasing in k in hazard rate ordering. Since the results can be transferred to the continuous-time case, and since the hazard rate ordering is stronger than the ordinary stochastic ordering, our results strengthen the corresponding results of Brown (1980). A sufficient condition for {Nk+m – Nk, k = 0, 1, ·· ·} to be non-increasing in k in hazard rate ordering as well as some sufficient conditions for the opposite monotonicity results are given. Finally, Brown's conjecture that DFR is necessary for concavity of the renewal function in the continuous-time case is discussed.

In this paper, we develop a unified approach for stochastic load balancing on various multiserver systems. We expand the four partial orderings defined in Marshall and Olkin, by defining a new ordering based on the set of functions that are symmetric, L-subadditive and convex in each variable. This new partial ordering is shown to be equivalent to the previous four orderings for comparing deterministic vectors but differs for random vectors. Sample-path criteria and a probability enumeration method for the new stochastic ordering are established and the ordering is applied to various fork-join queues, routing and scheduling problems. Our results generalize previous work and can be extended to multivariate stochastic majorization which includes tandem queues and queues with finite buffers.

If φ is a convex function and X a random variable then (by Jensen's inequality) ψ φ (X) = Eφ (X) – φ (EX) is non-negative and 0 iff either φ is linear in the range of X or X is degenerate. So if φ is not linear then ψ φ (X) is a measure of non-degeneracy of the random variable X. For φ (x) = x2, ψ φ (X) is simply the variance V(X) which is additive in the sense that V(X + Y) = V(X) + V(Y) if X and Y are uncorrelated. In this note it is shown that if φ ″(·) is monotone non-increasing then ψ φ is sub-additive for all (X, Y) such that EX ≧ 0, P(Y ≧ 0) = 1 and E(X | Y) = EX w.p.l, and is additive essentially only if φ is quadratic. Thus, it confirms the unique role of variance as a measure of non-degeneracy. An application to branching processes is also given.

In this paper we first prove an arrangement-decreasing property of partial sums of independent random variables when they are partially ordered through the likelihood ratio ordering. We then apply a similar argument to obtain a stochastic ordering of random processes via a comparison of their parameter functions, with special applications to Poisson and Wiener processes. Finally, in Section 4 we present some applications in reliability theory, queueing, and first-passage problems.

The bivariate characterization of stochastic ordering relations given by Shanthikumar and Yao (1991) is based on collections of bivariate functions g(x, y), where g(x, y) and g(y, x) satisfy certain properties. We give an alternate characterization based on collections of pairs of bivariate functions, g1(x, y) and g2(x, y), satisfying certain properties. This characterization allows us to extend results for single machine scheduling of jobs that are identical except for their processing times, to jobs that may have different costs associated with them.

Assume that we want to estimate – σ, the abscissa of convergence of the Laplace transform. We show that no non-parametric estimator of σ can converge at a faster rate than (log n)–1, where n is the sample size. An optimal convergence rate is achieved by an estimator of the form where xn = O(log n) and is the mean of the sample values overshooting xn. Under further parametric restrictions this (log n)–1 phenomenon is also illustrated by a weak convergence result.