# Acknowledgements

The author thanks all the members of the UniQuant community that took part to the Olympics of Mathematics on June $$23^{rd}$$ $$2015$$ and that afterwards discussed the results at the Quants’ Breakfast on July $$7^{th}$$ for useful insights and discussion.

# The Problem

The offices of two corporates (A and B) are housed in similar buildings, equipped with a set of smart elevators, that group employees before they take the elevator so as to optimize the average elevators’ usage rate, that is achieved by reaching a minimum in:

• the number of groups per minute;

• the service time (i.e. the time spent in the elevator by the employees).

At rush hour, however, the optimization software cannot do better than having three groups per minute with a service time of a minute and a half for each group (average values). Either of the buildings is served by a set of five elevator, but corporate B activates one more elevator at rush hour.
a) You are asked to compute the average waiting time of the elevator for the corporates’ employees at rush hour for both corporates.
Out of the peak hour, the optimization software is able to reduce the number of groups to only two per minute on average, while service time is kept constant so as to save power.
b) Would it be useful for corporate B to keep in service the additional elevator also in this case?

# Discussion

We discuss this problem in the context of the queueing theory Wiki: Queueing Theory, pionereed by Agner Krarup Erlang (1878 – 1929). (Erlang 1917)
In particular, we treat the queue in front of the elevators of the two buildings under the formalism of the M/M/c queue Wiki: M/M/c Queue, where:

1. the first M means that customer (i.e. a group of employees that takes the same elevator) arrival time is stochastic and follows a Poisson process;

2. the second M means that the service time (i.e. the time spent by an elevator to carry all the members of each group to the required floors and to get back at the ground) is stochastic and follows an exponential distribution;

3. the final c stems for the number of servers (i.e. the number of elevators available to employees).

Under the assumptions 1. and 2. above, the problem is made analytically tractable.

## The Distribution of Arrival Time

We introduce the count of arrival per minute $$k = 1,2,3, ...$$ where $$k$$ is incremented by one each time the optimization software creates a new group of employees, that will be carried by the same elevator. Probability of arrivals is: $\label{Poisson} P(k) = \frac{\lambda^k \cdot e^{- \lambda}}{k!}$ where $$\lambda$$ is the mean arrival count. The plot below shows the probability mass function (pmf) \eqref{Poisson} under the Poisson distribution in three cases ($$\lambda=1$$, $$\lambda=4$$ and $$\lambda=10$$).