Queues Applied to Telecoms

3Common Queues

Waiting doesn’t wait for anything.

Maurice Blanchot (1907–2003)

Queues are everywhere in our everyday life: in banks, at the post office, in traffic, at stock depots, on the telephone, at gas stations… They occupy an important place in human life ensuring that resources can be well-managed.

This chapter is dedicated to the study of queue theory. We elucidate mathematical models for these queues that can help improve their use in everyday life, and especially in the field of telecommunication where resources are very scarce.

3.1. Arrival process of customers in a queue

3.1.1. The Poisson process

As was explained in the first chapter, the Poisson process is important for describing the occurrence of random events. The arrival of customers in a queue can be modeled by a Poisson process. In this case, the two principal random variables to consider are:

• – the number of customers N(t) arriving in the queue during a length of time t. It is a positive, non-zero, integer-valued random variable: N(t) ∈ ℕ^*;
• – the time Τ that passes between two consecutive arrivals. We called this the inter-arrival time. It is a real, positive random variable: Τ ∈ ℝ⁺.

The two random variables N(t) and Τ are not independent: if the number of customers N(t) arriving in the queue is high, the inter-arrival time is short, and vice versa. Proposition 3.1 explains this relationship between these two random variables characterizing the arrival of customers in a queue.

PROPOSITION 3.1.– The ...