Mehmood_Yasir
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Actually I have not that much idea how to assess the accuracy of approximation, need to explore further, but I just developed a program that runs this process for 1 million time, interesting the empirical cdf of ##W_k## matches cdf of my approximation. would it be enough to verify the approximation.?Ray Vickson said:As I have indicated in post #48, the true situation is much more complicated than that, so your approach will give an approximation to the true answer. You would need to investigate further to assess the accuracy of the approximation.
On the other hand, you might be able to do something with an "equilibrium" analysis, where you look at the limiting distribution of wait times ##W_n## for customers with large ##n## For such customers far in the future, all the little details of entangled waiting times for customers ##1,2,3,\ldots ## will likely become unimportant, and the problem can be looked at much more simply. For example, you could look at the conditional distribution of waiting time for customer ##n##, given that customer ##n## is part of a group of ##k+1## customers who cross together. You could weight that by the probability of such a group, which you have found already.