By Yao H., Knessl Ch.
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Extra info for On the Shortest Queue Version of the Erlang Loss Model
Sample text
Knessl so we write (a + 1) S(L) (361) P¯ n(0) (L) = κ(ε) n! where κ is a constant and S(L) is an arbitrary function of L. To determine S and κ we use asymptotic matching to the expansion valid for m = L − O(1) and 0 < y < a. We thus expand (75) as y ↓ 0 and compare the result to the behavior of (361) as n → ∞. From (76) we get n 1 4a 2 + 1 − 1 (362) 2 and from (77) F (y) + log y → log (1 + a) as y → 0, so that F (y) ∼ −1/y and D(0) = F(y) = F(0) − y log y + y + y log(1 + a) + o(y). (363) Therefore, as y = εn → 0, e F(y)/ε ∼ e F(0)/ε (εn)−n en (1 + a)n .
From (84) we obtain √ 3/2 1/4 √ 2 4a + 1 + 1 4a 2 + 1 2a G 1 (x) ∼ √ √ 2 a−x 4a + 1 + 1 − 2a 1/4 4a 2 + 1 , √ 4a 2 + 1 − 2a a 3/2 =√ a−x and √ 2a 5/2 8a 2 + 2a + 1 1 G 1 (x) ∼ , x − X 0 (a) (4a + 1)3/2 x → a, (229) x → X 0 (a). (230) We note that G1 has singularities in both of these limits. We next consider the region L2 , where a − x = O(ε) (hence m = N − O(1)) and 0 < y < a. On this scale we set p(m, n) = P(x, y) = Q¯ L (y) (231) and (8) becomes (1 + ay − εL) Q¯ L (y) = Q¯ L (y − ε) + [a + ε(1 − L)] Q¯ L−1 (y) + (y + ε) Q¯ L (y + ε) (232) for L 0, with Q¯ −1 ≡ 0.
2 ∂y However from (141) we have exp( x ) ≡ 1/(1 − Aeτ ) and exp( ueτ ) so that the general solution of (161) has the form M=√ e−τ y) (161) = 1/(1 − L(u) J −1/2 . √ − A e−τ − u (162) This expresses M in terms of (u, τ ), up to an arbitrary function of u, and yields (40) up to (42). The function L(u) will be determined by asymptotic matching in Section 4. For the matching we need to know the behavior of J as (x, y) → (a, a). From (41) we see that J ∼ τ J0 (u), τ → 0, J0 (u) = 2a + 1 − a(2a + 1) 4a + 1−u (1 − u)2 (163) and hence the Jacobian vanishes near the upper corner, indicating a singularity in the mapping (43) between Cartesian and ray coordinates.
On the Shortest Queue Version of the Erlang Loss Model by Yao H., Knessl Ch.
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