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This article is devoted to the study of an M/G/1 queue with a particular vacation discipline. The server is due to take a vacation as soon as it has served exactly N customers since the end of the previous vacation. N may be either a constant or a random variable. If the system becomes empty before the server has served N customers, then it stays idle until the next customer arrival. Such a vacation discipline arises, for example, in production systems and in order picking in warehouses. We determine the joint transform of the length of a visit period and the number of customers in the system at the end of that period. We also derive the generating function of the number of customers at a random instant, and the Laplace–Stieltjes transform of the delay of a customer. © 2015 Wiley Periodicals, Inc. Naval Research Logistics, 2015

We present a group testing model for items characterized by marker random variables. An item is defined to be good (defective) if its marker is below (above) a given threshold. The items can be tested in groups; the goal is to obtain a prespecified number of good items by testing them in optimally sized groups. Besides this group size, the controller has to select a threshold value for the group marker sums, and the target number of groups which by the tests are classified to consist only of good items. These decision variables have to be chosen so as to minimize a cost function, which is a linear combination of the expected number of group tests and an expected penalty for missing the desired number of good items, subject to constraints on the probabilities of misclassifications. We treat two models of this kind: the first one is based on an infinite population size, whereas the second one deals with the case of a finite number of available items. All performance measures are derived in closed form; approximations are also given. Furthermore, we prove monotonicity properties of the components of the objective function and of the constraints. In several examples, we study (i) the dependence of the cost function on the decision variables and (ii) the dependence of the optimal values of the decision variables (group size, group marker threshold, and stopping rule for groups classified as clean) and of the target functionals (optimal expected number of tests, optimal expected penalty, and minimal expected cost) on the system parameters.© 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011

In this paper, we consider various queueing models in which the server can work at two different service speeds. The speed of the server depends on either the number of customers or the workload. Our main interest is in the model in which service speed adaptations can take place only at the arrival instants of an external Poisson observer. Using insightful probabilistic arguments, we give the structure of the steady-state queue length and workload distributions in the various models. In addition, in case the service speed can only be adapted right after departure instants based on the number of customers, we provide explicit and intuitively appealing expressions for the steady-state distribution of the number of customers present. © 2008 VVS.

We consider an M/M/1 queue with the special feature that the speed of the server alternates between two constant values sL and sH>sL. The high-speed periods are exponentially distributed, and the low-speed periods have a regularly varying distribution. We obtain explicit asymptotics for the tail of the workload distribution. The two cases in which the offered traffic load is smaller respectively larger than the low service speed are shown to result in completely different asymptotics.

We consider the M/M/1-queue and derive an explicit expression for the joint distribution of the number of arrivals and the number of departures in [0, t), given the number of customers initially present. The derivation is almost purely combinatorial, it avoids the use of generating functions, and immediately yields a simple probabilistic interpretation of the result.