In many stochastic models, particularly in queueing theory, Poisson arrivals both observe (see) a stochastic process and interact with it. In particular cases and/or under restrictive assumptions it has been shown that the fraction of arrivals that see the process in some state is equal to the fraction of time the process is in that state. In this paper, we present a proof of this result under one basic assumption: the process being observed cannot anticipate the future jumps of the Poisson process.
This paper considers tandem queues for which the order of performing service tasks can be changed, the service times being independent of this order. It determines the optimal order of service when either the service times of different tasks are nonoverlapping or, for two queues in tandem, the service time of one task is constant. For the optimal ordering, the waiting time of every customer is stochastically smaller than for any other.
Conditions are determined under which (a) the sum of forecasts of sales in market segments is preferred to a forecast of the whole market and (b) a forecast of an individual market segment is preferred to a forecast obtained through proration of a forecast of the whole market. The comparisons in (a) and (b) are also applied to the problem of forecasting for several time periods into the future.forecasting, statistics: time series
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. This content downloaded from 165.123.34.86 on Thu, 31 Dec 2015 03:46:04 UTC All use subject to JSTOR Terms and Conditions AbstractMoments of the delay distribution and other measures of performance for a multi-channel queue are shown to be bounded above by corresponding quantities for one or a collection of single-channel queues. UPPER BOUND; MULTI-CHANNEL OR SERVER; QUEUE Queues with several channels (servers) operating in parallel are sufficiently complex that bounds on measures of performance are of considerable interest. Brumelle [1], Kingman [3], and others have obtained bounds by comparing the performance of a multi-channel queue with one or a collection of single-channel queues that turn out to be 'better' or 'worse' in some sense. Both authors obtain the upper bound (38) in [3] for the expected delay in queue of a GI / G / c queue by involved but quite different arguments. However, Kingman does not stop there. He considers a 'modified' collection of c single-channel queues obtained from the original system by cyclically assigning the arrivals to the c channels so that every c th arrival is served at the same channel. He then remarks 'the modification clearly cannot have decreased the mean waiting time' and proceeds to write (39) in [3] and (11) below, an immediate consequence of this remark and a better bound than (38).Given the offhand manner in which the superior bound was obtained, its status has been in some doubt. Marchal [4] refers to [3] to support his conclusion that the stationary delay distribution is stochastically smaller in the original system than in the modified system. However, Marchal offers no additional argument to support this view. In the two-channel case, Piater [5] proves Kingman's remark is correct for expected delay but, contrary to Marchal, concludes that the stochastic ordering of the delay distributions can be false. However, this author does not find Piater's argument for his second conclusion convincing.In this paper, we explore a broader question: for what measures of performance is the original system better than any alternative which arbitrarily assigns arrivals to the channels? In the process, we find that Kingman's remark is true.
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