On the Erlang loss function
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ON THE ERLANG LOSS FUNCTION H. ALZER1,∗ and M. K. KWONG2 1
2
Morsbacher Straße 10, 51545 Waldbr¨ ol, Germany e-mail: [email protected]
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Hong Kong e-mail: [email protected] (Received May 18, 2019; revised March 1, 2020; accepted March 2, 2020)
Abstract. We present various properties of the Erlang loss function −1 ∞ e−at (1 + t)x dt (x ≥ 0, a > 0). B(x, a) = a 0
Among other results, we prove: (1) The function x �→ B(x, a)λ is convex on [0, ∞) for every a > 0 if and only if λ ≤ 0 or λ ≥ 1. (2) The function x �→ (1 − B(1/x, a))−1 is strictly convex on (0, ∞). This leads to the functional inequality 1 1 2 < + 1 − B(H(x, y), a) 1 − B(x, a) 1 − B(y, a)
(x, y > 0, x �= y),
where H(x, y) = 2xy/(x + y) denotes the harmonic mean of x and y. (3) Let a > 0. The inequality B(x, a) + B(1/x, a) ≤ 1 holds for all x > 0 if and only if a ≤ 1.
1. Introduction The function B(N, a) =
N aN aν −1 N ! ν=0 ν !
(0 ≤ N ∈ Z, a ∈ R)
is known as the Erlang loss function. It is named after the Danish mathematician and engineer Agner Krarup Erlang (1878-1929), who is called “the ∗ Corresponding
author. Key words and phrases: Erlang loss function, functional inequality, mean value, convexity, Laplace transform. Mathematics Subject Classification: 26E60, 33B20, 39B62, 44A10, 60K25, 90B20. c 2020 0236-5294/$ 20.00 © � 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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H. H. ALZER ALZER and and M. M. K. K. KWONG KWONG
father of teletraffic theory” (Medhi [7, p. 309]). In 1917, Erlang showed that B(N, a) is “the probability that a call, which is a member of a Poisson stream of parameter a, arriving at a group of N telephone trunks will be rejected” (Jagerman [4, p. 525]). In view of its importance in solving telecommunication problems, the Erlang loss function attracted the attention of many researchers, who discovered numerous properties of B(N, a). For detailed information on this subject we refer to Giambene’s monograph [3] and to Medhi’s survey article [7]. Jagerman [4] extended the definition of B(N, a) to complex numbers by using the integral formula −1 ∞ −at x e (1 + t) dt . B(x, a) = a 0
This function can be expressed in terms of the incomplete gamma function. Indeed, we have the representation B(x, a) =
e−a ax . Γ(x + 1, a)
The elegant formulae x ∂ B(x, a) = − 1 B(x, a) + B(x, a)2 ∂a a and (1.1)
B(x + 1, a)−1 =
x+1 B(x, a)−1 + 1 a
are given in [4] and [7], respectively. The function x �→ B(x, a) (a > 0) is strictly decreasing on [0, ∞) with B(0, a) = 1 and limx→∞ B(x, a) = 0. Jagers and van Doorn [5] proved that (1.2)
B ′′ (x, a) > 0
and (log B(x, a))′′ < 0
(x ≥ 0, a > 0).
(Here and in what follows, a prime indicates differentiation with respect to the first variable.) From (1.2) we conclude that x �→ B(x, a) is strictly convex and that x �→ 1/B(x, a) is strictly log-convex on [0, ∞) for every a > 0. The log-convexity was also proved by Jagerman [4]. Using (1.2) and Je
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