I think this question belongs to queue-theory but it doesn't really mention anything about queue-theory:
Suppose we have a queue of jobs serviced by one server. There is a total of $n$ jobs in the system. At time $t$, each remaining job independently decides to join the queue to be serviced with probability $p = d/n$, where $d < 1$ is a constant. Each job has a processing time of $1$ and at each time the server services one job, if the queue is nonempty. Show that with high probability, no job waits more than $\Omega(\ln n)$ time to be serviced once it joins the queue.
I really have no idea how to attack this one, I do not wish for an exact answer but rather someone to point me in the right direction, I know this has something to be with the number $e$ powered to something which reminds me to some probability distribution functions but I'm not sure how to proceed here. Thanks in advance!
Conditionally fixing the distribution of the waiting time - P.35: http://www.win.tue.nl/~iadan/queueing.pdf
See from this equation, reading backwards, for less of a hint and more of an answer:
$$P(W\gt t| W \gt 0) = \frac {P(W \gt t)} {P(W \gt 0)} = e^{-u (1-p)t} $$
$W$ is with probability $(1−ρ)$ equal to zero, and with probability $ρ$ equal to an exponential random variable with parameter $µ(1 − ρ)$. So the conditional waiting time $W|W > 0$ is exponentially distributed with parameter $µ(1−ρ)$.
The exponential distribution (also known as negative exponential distribution) inhibits infinite divisibility and is the probability distribution that describes the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being a memoryless.
Kendall's notation (or sometimes Kendall notation) is the standard system used to describe and classify a queueing node.
Additional reading: Wikipedia queuing, ruin, maximum entropy probability distribution, even Pollaczek–Khinchine formula for claims with expotential distribution - derivation.
Other documents showing the usage of $ e^{-u (1-p)t} $
Waiting Systems - Page 9: https://www.netlab.tkk.fi/opetus/s383143/kalvot/E_mm1jono.pdf
ETSN10 Formula Sheet - Page 2: http://www.eit.lth.se/fileadmin/eit/courses/etsn10/formula_sheet.pdf
Introduction to Queueing theory - Computer Communication Networks - Page 14: http://web.iitd.ac.in/~jbseo/ell785_16/Queueing_theory_2016.pdf
ST333 Applied Stochastic Processes: Outline ... - University of Warwick - Page: 66: http://web.warwick.ac.uk/statsdept/teaching/notes/st333.pdf