Talk:Arrival theorem

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Is this a theorem?[edit]

Is this a theorem? ...or is it just a statement of some part of the assumptions of the model? The latter seems the case to me, and is possibly indicated by the alternative names being "properties". How many sources call this a "theorem"? The article indicates that at least one does, but I don't have access. Melcombe (talk) 09:43, 3 June 2009 (UTC)[reply]

I found naming this article tricky as all the terms listed in the opening sentence are used by different authors to refer to the same concept. Searching on Google Scholar and Google Books suggested to me that arrival theorem was commonplace and would apply to all the results, but I agree that on reflexion 'property' is probably a more apt term. (Most of the references are available on Google Books.) Gareth Jones (talk) 11:41, 3 June 2009 (UTC)[reply]

I see from the Asmussen book that the cases covered are rather more complicated than I had thought, so "theorem" is appropriate even if it is a moderately straightforward deduction in some cases. And it dores seem that "arrival theorem" is used in the literature I spottted some content in "Introduction to probability models", by Sheldon M. Ross (9'th edition,Academic Press, 2007, ISBN 0125980620) which might be a resonably accessible reference to some of this. However, I think some clarification of the present content is needed ... in particular, it would be good to have an indication of the types of queue to which the "Arrival theorem" does not apply. Melcombe (talk) 13:17, 3 June 2009 (UTC)[reply]

Yes, good point. I'm aware of Boucherie's On the arrival theorem for product form queueing networks with blocking (Performance Evaluation, Volume 29, Issue 3, Pages 155-222 (April 1997)) which demonstrates the arrival theorem in a more general context than that currently given in the Wikipedia article, and shows how the theorem does not apply to all Kingman networks. There's definitely scope for expansion of this article, I'll try and find some time to work on it soon. Gareth Jones (talk) 22:56, 3 June 2009 (UTC)[reply]

I raised the point because I saw, in the response to a Google query, things like "the arrival theorem does not apply to systems where queues have a finite capacity". By starting with thoughts of "Kingman networks" you may be (either or both) assuming too much knowledge of the reader or putting to strong a limit on classes of queues being considered. Melcombe (talk) 10:34, 4 June 2009 (UTC)[reply]

Arrival theorem citation[edit]

The paper to cite for the arrival theorem should be, in addition to Sevcik & Mitrani, this one: Stationary state probabilities at arrival instants for closed queueing networks with multiple types of customers Steve S Lavenberg, Martin Reiser. Journal of Applied Probability, 1048--1061, JSTOR, 1980

The cited paper develops the MVA algorithm, but it does so using normalizing constants manipulations and the product-form formulas, it does not look at the process embedded at arrival or departure instants.