We establish some general structural results and derive some simple formulas describing the time-dependent performance of the Mt/G/∞ queue (with a nonhomogeneous Poisson arrival process). We know that, for appropriate initial conditions, the number of busy servers at time t has a Poisson distribution for each t. Our results show how the time-dependent mean function m depends on the time-dependent arrival-rate function λ and the service-time distribution. For example, when λ is quadratic, the mean m(t) coincides with the pointwise stationary approximation λ(t)E[S], where S is a service time, except for a time lag and a space shift. It is significant that the well known insensitivity property of the stationary M/G/∞ model does not hold for the nonstationary Mt/G/∞ model; the time-dependent mean function m depends on the service-time distribution beyond its mean. The service-time stationary-excess distribution plays an important role. When λ is decreasing before time t, m(t) is increasing in the service-time variability, but when λ is increasing before time t, m(t) is decreasing in service-time variability. We suggest using these infinite-server results to approximately describe the time-dependent behavior of multiserver systems in which some arrivals are lost or delayed.
Networks are critical to modern society, and a thorough understanding of how they behave is crucial to their efficient operation. Fortunately, data on networks is plentiful; by visualizing this data, it is possible to greatly improve our understanding. Our focus is on visualizing the data associated with a network and not on simply visualizing the structure of the network itself. We begin with three static network displays; two of these use geographical relationships, while the third is a matrix arrangement that gives equal emphasis to all network links. Static displays can be swamped with large amounts of data; hence we introduce directmanipulation techniques that permit the graphs to continue to reveal relationships in the context of much more data. In effect, the static displays are parameterized so that interesting views may easily be discovered interactively. The software to carry out this network visualization is called SeeNet.
In this paper we describe the mean number of busy servers as a function of time in an Mt/G/∞ queue (having a nonhomogeneous Poisson arrival process) with a sinusoidal arrival rate function. For an Mt/G/∞ model with appropriate initial conditions, it is known that the number of busy servers at time t has a Poisson distribution for each t, so that the full distribution is characterized by its mean. Our formulas show how the peak congestion lags behind the peak arrival rate and how much less is the range of congestion than the range of offered load. The simple formulas can also be regarded as consequences of linear system theory, because the mean function can be regarded as the image of a linear operator applied to the arrival rate function. We also investigate the quality of various approximations for the mean number of busy servers such as the pointwise stationary approximation and several polynomial approximations. Finally, we apply the results for sinusoidal arrival rate functions to treat general periodic arrival rate functions using Fourier series. These results are intended to provide a better understanding of the behavior of the Mt/G/∞ model and related Mt/G/s/r models where some customers are lost or delayed.
We compare balanced randomization with four adaptive treatment allocation procedures in a clinical trial involving two treatments. The objective is to treat as many patients in and out of the trial as effectively as possible. Randomization is a satisfactory solution to the decision problem when the disease in question is at least moderately common. Adaptive procedures are more difficult to use, but might play a role in clinical research when a substantial proportion of all patients with the disease are included in the trial.
Synchronous relaxation, a new, general-purpose, efficient method for parallel simulation, is proposed, The method is applied to obtain a new parallel algorithm for simulating large circuit-switched communication networks. To show that synchronous-relaxation method is efficient, we present the results of circuit-switched network simulation experiments, and analytic approximations derived from a mathematical model of the simulation method.
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