Measuring the Overlap with Other Customers in the Single Server Queue

The single server queue is one of the most basic queueing systems to model stochastic waiting dynamics. Most work involving the single server queue only analyzes the customer or agent behavior. However, in this work, we are inspired by COVID-19 applications and are interested in the interaction between customers and more specifically, the time that adjacent customers overlap in the queue. To this end, we derive a new recursion for this overlap time and study the steady state behavior of the overlap time via simulation and probabilistic analysis. We find that the overlap time between adjacent customers in the M/M/1 queue has a conditional distribution that is given by an exponential distribution, however, as the distance between customers grows, the probability of a non-negative overlap time decreases geometrically. We also find via simulation that the exponential distribution still holds when the distribution is non-exponential, hinting at a more general result.