A note on Erlang’s formulas

A generalisation of erlang's formulas in queueing theory

“…In the last sections we consider anew a system with infinitely many lines. This was the subject of our note [6]. Unfortunately, the final formulas of [6J are vitiated by a gross error which slipped into our reasoning*.…”

On the rate of convergence for infinite server Erlang–Sevastyanov’s problem

“…Substitution of this into (44) shows that, indeed, (39) implies (38), under the assumption (40), the latter being guaranteed by (36).…”

“…Khinchin's works in [30, The Editor's Introduction], the paper was, in fact, fully prepared to publication in 1954-1955. Earlier, the original Erlang's formulae with exponential service time distribution were extended on systems with an infinite number of servers [28], and later (1965) this result was tackled by a different method in [38]. Among all these results, [45] remains the most advanced achievement in that period.…”

“…Then we wait again until the "next" momentτ 0,R and repeat the whole procedure of the "attempt" to meet both components at ∆ 0 . Thus, we will evaluatē τ 0 by means of some geometric series, which would guarantee the desired inequality (38). Hence, let us show the bound (39) first.…”

“…s) kL m,a−1/m (X s , Y s ) ds ≤ CL m,a+ℓ/m (x, y).Due to Fatous's lemma, with ℓ = k + δ (i.e., ℓ > k) this imples,E x,yLm,a,k (τ 0,R , Xτ 0,R , Yτ 0,R ) + C ′ E x,y τ 0,R 0 s) kL m,a−1/m (X s , Y s ) ds ≤ CL m,a+ℓ/m (x, y).SinceL m,a−1/m (X s ) ≥ 1 on s <τ 0 (and on s <τ 0,R ), we getE x,yτ k+1 0,R ≤ CL m,a+ℓ/m (x, y),so that (39) is established. Now let us show(38). As explained above, to this aim we choose t 1 so thatsup u: L m,a−1/m (u)≤R+1 P u (τ 0 > t 1 ) ≤ t −(k+1) 1 sup u: L m,a−1/m (u)≤R+1…”