On the limiting probabilities of the M/Er/1M/Er/1 queueing system

• A new recurrent formula to calculate the M/Er/1M/Er/1 queue distribution is presented.
• The new recurrent formula is numerically stable and very efficient.
• A new analytical formula for the queue distribution is also presented.
• A probabilistic interpretation for each of the analytical formula terms is given.

In this paper, two new formulations of the steady-state probability distribution of the number of customers in an M/Er/1M/Er/1 queue system are presented. On one hand, a new recurrent method which is numerically stable and computationally more efficient than the existing ones, in terms of memory and time, is presented. On the other hand, a new analytic formula, simpler than the current ones, which additionally has associated a probabilistic interpretation of its terms, that gives a new insight about this queue and its behavior, is also presented.