Talk:Birth–death process

Infinite Calling Population Assumption

The article says "In queueing theory the birth-death process is the most fundamental example of a queueing model, the M/M/C/K/$\infty$/FIF0". Why is it that the calling population must be infinite? It seems that a finite calling population would be easy to model with a birth-death process since the $\mu ,\lambda$ parameters can depend on the number of customers in the queue, which is easily related to the number of customers not in the queue when the population is finite. A5 15:24, 1 June 2007 (UTC)

If you have a finite calling population, rather than infinite one, you can have constant parameters $\mu ,\lambda$ for all levels of the queue, but the blocking probability changes. Have a look at Erlang unit, there is a parameter $A=\lambda /\mu$ which is constant reguardless of the number of queueing customers, however the blocking probability changes for the infinite population (Erlang C formula) in the finite case (Engset formula), this make it computationally simpler then having to place a weight on $\lambda$ as the queue increases. Aiden Fisher 06:12, 5 June 2007 (UTC)

The default assumption in both Birth-Death and Queueing models is that the population is infinite. Indeed A5 is correct that the assumption need not be the case in either. However, with 'sufficiently large' populations, assuming an infinite population doesn't reduce the accuracy of analytical investigations of the process but does make such investigations much more tractable - that is, it makes such investigations both easier to conduct and, when numbers are involved, the computations of the flow equations are much simpler and fast. Largely for this reason, the predictive power rather than the exact match to reality, the M/M/C/K/$\infty$/FIFO queue is the most fundamental queueing model used. I think that the sentence should be clarified to something like, "The birth-death process is a fundamental building block of the most widely used queueing model, the M/M/C/K/$\infty$/FIFO queue." Margaretpierson (talk) 22:43, 9 September 2012 (UTC)

With respect to your discussion of how to model a finite pool of users when a 'small' pool of potential callers exists, i.e., when an approximation with an infinite population is inappropriate, I agree with Aiden Fisher that you'd likely be better off just using the Birth-Death process directly.

For general consumption: If you have a population so small that, for instance, the number of people in line impacts the arrival rate of new 'customers' then it is poorly fit with an M/M/c/K/</math>\infty</math> model. Suppose you are looking at the failure rate of light bulbs in your home. If several are blown out while you're replacing the first one, you will see a slow down in the rate of light bulb failure because many lamps are "in line" for service. As A5 notes, the model must be modified to account for the fact that the number in queue will impact the arrival rate. Margaretpierson (talk) 22:41, 9 September 2012 (UTC)

Image

If this is supposed to represent the size of a population, should we just make lambda 0 be equal to 0 since it would be an absorbing state? Brusegadi (talk) 05:41, 13 August 2008 (UTC)

Note of deletion

I have deleted the recent addition of Dec 30 which had all the missing and misformatted equations. It seems to have been copied from http://staff.um.edu.mt/jskl1/simweb/mm1.htm which may violate copyright. Even if not, the copyng stage seemed to have resulted in many errors and would need to be done more carefully. I note the same addition was attempted in both Birth-death process and M/M/1 model. Melcombe (talk) 09:52, 13 January 2010 (UTC)