On the Convergence of Multiclass Queueing Networks in Heavy Traffic
On the Convergence of Multiclass Queueing Networks in Heavy Traffic
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3 from Iglehart and Whitt [20] yields v^'"<"' 0. D. Lemma 4. 3 Suppose the convergence tn (3. 1) holds. Then f"(f) — e< u. O. C, where e is the d-dimensional vector of ones. Proof. Let iy, "(t) = iPV"(nt). Then, Kif"{t)) - -^]{nt) = -T^int) < rv;(f;(t)), Because t"{s) < s for s > 0, 1 With the assumption of (3. 7) and Lemma 4. 2, the lemma is proved. Lemma 4. 4 Suppose the convergence in (3-7) holds. Then, f"(0 — W'*(<) u. O. C. As n — 00. 9 Proof. Because w^ir^^it)) - €^(t) = t - r;(f ) < n-;(r;(o), we have rv^;(f;(0) - ^s"("*) = ^;(0 < K(f^{t)). The lemma follows immediately from assumption (37) and Lemmas 4. 2 and 43. □ Lemma 4. 5 Suppose the convergence tn (3. 7) holds. Then i"(<) — A< u. O. C. Proof. It follows from (23) that C k=i where E"(0 = ^F"(nf), £)"(/) = ^D^'int) and $''"•"(0 = ]^4>^{[nt]) hv /t = 1 c. Therefore. (4. 7) A''{t)-Xt = E'^(0-af + Xl(''"(^, "(/))-P(D^(0) fc=i + P'(D"(i) - AC'f"(<)) - P'XC'ite - f"(<)), where we have used the fact that A = a+P'A, and Pk denotes the k^^ row of P.
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To quickly assess the difficulty of the text, read a short excerpt:
3 from Iglehart and Whitt [20] yields v^'"<"' 0. D. Lemma 4. 3 Suppose the convergence tn (3. 1) holds. Then f"(f) — e< u. O. C, where e is the d-dimensional vector of ones. Proof. Let iy, "(t) = iPV"(nt). Then, Kif"{t)) - -^]{nt) = -T^int) < rv;(f;(t)), Because t"{s) < s for s > 0, 1 With the assumption of (3. 7) and Lemma 4. 2, the lemma is proved. Lemma 4. 4 Suppose the convergence in (3-7) holds. Then, f"(0 — W'*(<) u. O. C. As n — 00. 9 Proof. Because w^ir^^it)) - €^(t) = t - r;(f ) < n-;(r;(o), we have rv^;(f;(0) - ^s"("*) = ^;(0 < K(f^{t)). The lemma follows immediately from assumption (37) and Lemmas 4. 2 and 43. □ Lemma 4. 5 Suppose the convergence tn (3. 7) holds. Then i"(<) — A< u. O. C. Proof. It follows from (23) that C k=i where E"(0 = ^F"(nf), £)"(/) = ^D^'int) and $''"•"(0 = ]^4>^{[nt]) hv /t = 1 c. Therefore. (4. 7) A''{t)-Xt = E'^(0-af + Xl(
What to read after On the Convergence of Multiclass Queueing Networks in Heavy Traffic?
You can find similar books in the "Read Also" column, or choose other free books by J G Jiangang Dai to read online
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