We consider the problem of selecting a stopping time τ which determines when to exit an investment project when the project's cumulative profit up to time t is Xt, where {Xt : t ≥ 0} is a Brownian motion with drift μ and variance σ2. The profit rate μ never changes over time, but μ is not directly observable. Specifically, μ takes the value μH > 0 when in the high state and μL < 0 when in the low state, and the initial probability p0 that the project is in the high state is known. The decision-maker seeks to maximize the expected discounted profit up to time τ. Using the theory of stochastic differential equations, we show that it is optimal to exit only when the posterior probability Pt of being in the high state falls below a critical number p*, and we produce a simple, closed form for p*. Our most surprising comparative-statics result is that the expected discounted profit increases with |μL|, provided |μL| is large.

For a compound Poisson process (CPP) with only positive jumps, an elegant formula connects the density of the hitting time for a lower straight line with that of the process itself at time t, h(x; t), considered as a function of time and position jointly. We prove an analogous (albeit more complicated) result for the first time the CPP crosses an upper straight line. We also consider the conditional density of the CPP at time t, given that the upper line has not been reached before t. Finally, it is shown how to compute certain moment integrals of h.

We study a stochastic scheduling problem with a single machine subject to random breakdowns. We address the preemptive-repeat model; that is, if a breakdown occurs during the processing of a job, the work done on this job is completely lost and the job has to be processed from the beginning when the machine resumes its work. The objective is to complete all jobs so that the the expected weighted flow time is minimized. Limited results have been published in the literature on this problem, all with the assumption that the machine uptimes are exponentially distributed. This article generalizes the study to allow that (1) the uptimes and downtimes of the machine follow general probability distributions, (2) the breakdown patterns of the machine may be affected by the job being processed and are thus job dependent; (3) the processing times of the jobs are random variables following arbitrary distributions, and (4) after a breakdown, the processing time of a job may either remain a same but unknown amount, or be resampled according to its probability distribution. We derive the necessary and sufficient condition that ensures the problem with the flow-time criterion to be well posed under the preemptive-repeat breakdown model. We then develop an index policy that is optimal for the problem. Several important situations are further considered and their optimal solutions are obtained.

We consider a single-server queue with exponential service time and two types of arrivals: positive and negative. Positive customers are regular ones who form a queue and a negative arrival has the effect of removing a positive customer in the system. In many applications, it might be more appropriate to assume the dependence between positive arrival and negative arrival. In order to reflect the dependence, we assume that the positive arrivals and negative arrivals are governed by a finite-state Markov chain with two absorbing states, say 0 and 0′. The epoch of absorption to the states 0 and 0′ corresponds to an arrival of positive and negative customers, respectively. The Markov chain is then instantly restarted in a transient state, where the selection of the new state is allowed to depend on the state from which absorption occurred.

The Laplace–Stieltjes transforms (LSTs) of the sojourn time distribution of a customer, jointly with the probability that the customer completes his service without being removed, are derived under the combinations of service disciplines FCFS and LCFS and the removal strategies RCE and RCH. The service distribution of phase type is also considered.

It is proved that the total average waiting time is a convex function of the routing densities for regular routing to parallel queues.

In the present article, we develop some efficient bounds for the distribution function of a two-dimensional scan statistic defined on a (double) sequence of independent and identically distributed (i.i.d.) binary trials. The methodology employed here takes advantage of the connection between the scan statistic problem and an equivalent reliability structure and exploits appropriate techniques of reliability theory to establish tractable bounds for the distribution of the statistic of interest. An asymptotic result is established and a numerical study is carried out to investigate the efficiency of the suggested bounds.

We analyze a feedback system consisting of a finite buffer fluid queue and a responsive source. The source alternates between silence periods and active periods. At random epochs of times, the source becomes ready to send a burst of fluid. The length of the bursts (length of the active periods) are independent and identically distributed with some general distribution. The queue employs a threshold discarding policy in the sense that only those bursts at whose commencement epoch (the instant at which the source is ready to send) the workload (i.e., the amount of fluid in the buffer) is less than some preset threshold are accepted. If the burst is rejected then the source backs off from sending. We work within the framework of Poisson counter-driven stochastic differential equations and obtain the moment generating function and hence the probability density function of the stationary workload process. We then comment on the stability of this fluid queue. Our explicit characterizations will further provide useful insights and “engineering” guidelines for better network designing.

It is shown that on fairly weak conditions, the current solutions of a metaheuristic following the ant colony optimization paradigm, the graph-based ant system, converge with a probability that can be made arbitrarily close to unity to one element of the set of optimal solutions. The result generalizes a previous result by removing the very restrictive condition that both the optimal solution and its encoding are unique (this generalization makes the proof distinctly more difficult) and by allowing a wide class of implementation variants in the first phase of the algorithm. In this way, the range of application of the convergence result is considerably extended.

We prove that no stochastic domination exists between the effective resistance of a spherically symmetric random tree and that of a branching process in a varying environments tree if they grow according to the same law of distribution.