In the case of a queueing model, it is very likely that service time distributions in a real life situation, do not have an exponential tail. This means that all the analytic solutions derived in any standard textbook are no longer applicable. If the server following a generic distribution, the expression linking the distributions such as Waiting time distribution, First passage time distributions, etc. and service time distributions is in the “Laplace Transform space”.

Laplace transform is a useful mathematical tool that one must be familiar with, while doing applied work. It is widely used in Queueing models where probability distributions are characterized in terms of transforms. Inverting a Laplace transform to get to the probability distribution is an essential task in Queueing theory. For textbook examples and simple Markovian models, one might be fortunate to find convenient forms for LT inversion. However for most of the real life situations, a practitioner needs to know a way to numerically invert LT.

Via TES magazine Mary Pat Wenderoth stops herself mid-lesson and asks her class a question about the day’s work. The students turn to their notes but she stops them. “Don’t look it up. Imagine your brain is a forest and your memory is in there somewhere. The more times you make a path to that memory, the stronger that path becomes. Try to figure it out.” Wenderoth is a principal lecturer in biology at the University of Washington in Seattle, US.

Parrondo’s Paradox : A combination of losing strategies becomes a winning strategy. This paradox can be seen via a simple simulation of three games. Game Type 1 – You flip a biased coin that has 1/2-epsilon as the probability of heads. For each head you get $1 and for each tail you lose $1 Game Type 2 – If your capital is a multiple of 3, you flip a biased coin that has 1/10-epsilon, as probability of heads.

Computing steady state probabilities for a queueing system is somewhat easier than computing the transient distributions. The latter typically comprises a differential-difference equation and the usual trick of recursive substitution fails as there is a derivative in the equation. The tools employed in solving a differential-difference equation are Generating functions, Laplace transforms and PDE solving tricks. Only for simple systems such as M/M/1 can one slog out and find a closed form solution.