There is a lot of literature on this topic. For instance, consider M/M/1 queues, and define ρ as the traffic utilisation factor or traffic intensity (ρ = λ/μ). You can then follow the derivation of L, the average queue length, here or in most queuing theory textbooks:
L = ρ / (1 - ρ)
As ρ → 1 (i.e., λ → μ), the denominator (1 - ρ) approaches 0, indicating that the limit diverges. So, as the arrival rate approaches the service rate, the system reaches a point where the queue length is unbounded and grows indefinitely over time. This implies that, practically, the queue will continue to grow without limit, leading to an infinite average queue length.
Its worth noting that this is primarly due to the stochastic nature of the arrival and service processes. Even when the average arrival rate equals the average service rate, the randomness in arrival times and service times causes periods where arrivals temporarily outpace services, leading to a buildup in the queue. Because the system lacks the ability to dissipate this buildup (as it would if λ<μ), the queue length tends to grow without bound over time.
