This paper is devoted to the performance analysis and optimization of blood testing procedures. We present a queueing model of two queues in series, representing the two stages of a blood-testing procedure. Service (testing) in stage 1 is performed in batches, whereas it is done individually in stage 2. Since particular elements of blood can only be stored and used within a finite time window, the sojourn time of blood units in the system of two queues in series is an important performance measure, which we study in detail. We also introduce a profit objective function, taking into account blood acquisition and screening costs as well as profits for blood units, which were found uncontaminated and were tested fast enough. We optimize that profit objective function w.r.t. the batch size and the length of the time window.

In this paper we derive limit theorems for the conditional distribution of X1 given Sn=sn as n→ ∞, where the Xi are independent and identically distributed (i.i.d.) random variables, Sn=X1+··· +Xn, and sn/n converges or sn ≡ s is constant. We obtain convergence in total variation of PX1∣ Sn/n=s to a distribution associated to that of X1 and of PnX1∣ Sn=s to a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.

Hazard rates play an important role in various areas, e.g. reliability theory, survival analysis, biostatistics, queueing theory, and actuarial studies. Mixtures of distributions are also of great preeminence in such areas as most populations of components are indeed heterogeneous. In this study we present a sufficient condition for mixtures of two elements of the same natural exponential family (NEF) to have an increasing hazard rate. We then apply this condition to some classical NEFs having either quadratic or cubic variance functions (VFs) and others as well. Particular attention is paid to the hyperbolic cosine NEF having a quadratic VF, the Ressel NEF having a cubic VF, and the NEF generated by Kummer distributions of type 2. The application of such a sufficient condition is quite intricate and cumbersome, in particular when applied to the latter three NEFs. Various lemmas and propositions are needed to verify this condition for such NEFs. It should be pointed out, however, that our results are mainly applied to a mixture of two populations.

Let f be a probability density function on (a, b) ⊂ (0, ∞), and consider the class Cf of all probability density functions of the form Pf, where P is a polynomial. Assume that if X has its density in Cf then the equilibrium probability density x ↦ P(X > x) / E(X) also belongs to Cf: this happens, for instance, when f(x) = Ce−λx or f(x) = C(b − x)λ−1. We show in the present paper that these two cases are the only possibilities. This surprising result is achieved with an unusual tool in renewal theory, by using ideals of polynomials.

We consider a dam model with a compound Poisson input having positive jumps in which the release rule can be dynamically controlled between two release functions, r1 and r2. The dam process has two switchover levels, a and b, 0 < a < b < ∞, which are prescribed to switch the release rule from r2 to r1 only when the dam downcrosses levels and switch from r1 to r2 only when the dam upcrosses level b. We derive integral equations whose solutions determine the stationary distribution of the dam content and demonstrate such a determination explicitly for the case of exponential jumps. We also compute an expression for the average number of switches from r1 to r2, and vice versa; consider an approximation to the general case (with b − a being sufficiently large with respect to r1 and r2 and the jump sizes), which allows the use of asymptotic results from renewal theory; and study the wet period analysis for the case of exponential jumps.

We consider control policies for perishable inventory systems with random input whose purpose is to mitigate the effects of unavailability. In the basic uncontrolled system, the arrival times of the items to be stored and the ones of the demands for those items form independent Poisson processes. The shelf lifetime of every item is finite and deterministic. Every demand is for a single item and is satisfied by the oldest item on the shelf, if available. The first controlled model excludes the possibility of unsatisfied demands by introducing a second source of fresh items that is completely reliable and delivers without delay whenever the system becomes empty. In the second model, there is no additional ordering option by outsourcing. However, to avoid the most adverse effects of unavailability, the demands are classified into different categories of urgency. An incoming demand is satisfied or not according to its category and the current state of the system. For both models, we determine the steady-state distribution of the virtual outdating process, which is then used to derive the relevant cost functionals: the steady-state distribution and expected value of the number of items in the system, the rate of outdatings, as well as, for model 1, the rate of special orders from the external source and, for model 2, the rate of unsatisfied demands.

We studied several group testing models with and without processing times. The objective was to choose an optimal group size for pooled screening of a contaminated population so as to collect a prespecified number of good items from it with minimum testing expenditures. The tested groups that were found to be contaminated were used as a new sampling population in later stages of the procedures. Since testing may be time-consuming, we also considered deadlines to be met for the testing process. We derived algorithms and exact results for the underlying distributions, enabling us to find optimal procedures. Several numerical examples are given.