Optimality of the Second Order Metrics of the M/M/2 System with Heterogeneous Service Rates

Author(s):  
X. Liu
Author(s):  
David Gamarnik ◽  
John N. Tsitsiklis ◽  
Martin Zubeldia

We consider a heterogeneous distributed service system consisting of n servers with unknown and possibly different processing rates. Jobs with unit mean arrive as a renewal process of rate proportional to n and are immediately dispatched to one of several queues associated with the servers. We assume that the dispatching decisions are made by a central dispatcher with the ability to exchange messages with the servers and endowed with a finite memory used to store information from one decision epoch to the next, about the current state of the queues and about the service rates of the servers. We study the fundamental resource requirements (memory bits and message exchange rate) in order for a dispatching policy to be always stable. First, we present a policy that is always stable while using a positive (but arbitrarily small) message rate and [Formula: see text] bits of memory. Second, we show that within a certain broad class of policies, a dispatching policy that exchanges [Formula: see text] messages per unit of time, and with [Formula: see text] bits of memory, cannot be always stable.


2010 ◽  
Vol 24 (4) ◽  
pp. 473-483
Author(s):  
Jun Li ◽  
Yiqiang Q. Zhao

In this article, we consider the two-node fork-join model with a Poisson arrival process and exponential service times of heterogeneous service rates. Using a mapping from the queue lengths in the parallel nodes to the join queue length, we first derive the probability distribution function of the join queue length in terms of joint probabilities in the parallel nodes and then study the exact tail asymptotics of the join queue length distribution. Although the asymptotics of the joint distribution of the queue lengths in the parallel nodes have three types of characterizations, our results show that the asymptotics of the join queue length distribution are characterized by two scenarios: (1) an exact geometric decay and (2) a geometric decay with the prefactor n−1/2.


Author(s):  
W. L. Bell

Disappearance voltages for second order reflections can be determined experimentally in a variety of ways. The more subjective methods, such as Kikuchi line disappearance and bend contour imaging, involve comparing a series of diffraction patterns or micrographs taken at intervals throughout the disappearance range and selecting that voltage which gives the strongest disappearance effect. The estimated accuracies of these methods are both to within 10 kV, or about 2-4%, of the true disappearance voltage, which is quite sufficient for using these voltages in further calculations. However, it is the necessity of determining this information by comparisons of exposed plates rather than while operating the microscope that detracts from the immediate usefulness of these methods if there is reason to perform experiments at an unknown disappearance voltage.The convergent beam technique for determining the disappearance voltage has been found to be a highly objective method when it is applicable, i.e. when reasonable crystal perfection exists and an area of uniform thickness can be found. The criterion for determining this voltage is that the central maximum disappear from the rocking curve for the second order spot.


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