pfqn_xia.h File Reference

Xia's asymptotic approximation of the normalizing constant of a load-dependent (multiserver) closed network. More...

#include <cmath>
#include <cstddef>
#include <vector>
#include "line/api/pfqn/pfqn_asympt_common.h"
#include "line/num/number.h"
#include "line/util/error.h"
#include "line/util/matrix.h"

Include dependency graph for pfqn_xia.h:

Go to the source code of this file.

Namespaces
namespace	line
namespace	line::pfqn

Functions
template<class T>
T	line::pfqn::pfqn_xia (const std::vector< T > &L, int N, const std::vector< T > &s)
	Xia's asymptotic approximation of the normalizing constant of a load-dependent (multiserver) closed network.

Detailed Description

Xia's asymptotic approximation of the normalizing constant of a load-dependent (multiserver) closed network.

Templated port of jar/src/main/java/jline/api/pfqn/ld/Pfqn_xia.java. MATLAB has no counterpart.

The demands are first rescaled so that the largest per-server utilization rho_i = L_i / s_i is one. The stations that attain it are the BOTTLENECK SET B; they saturate and contribute the M/M/s saturated term, while every other station contributes its finite-capacity Erlang-like partial sum

F(u, k) = sum_{j < k} u^j / j! + (u^k / k!) / (1 - u/k),

the closed form of the geometric tail beyond the k-th server. The result is

log G ~ -log((B-1)!) - N log(c) + sum_{b in B} [ s_b log L_b - log(s_b!) ]

• sum_{k not in B} log F(L_k, s_k),

c being the rescaling factor. Note the leading behaviour in N enters ONLY through -N log c: this is the large-population limit, so the approximation does not resolve the O(1) corrections that a finite population carries.

A NON-BOTTLENECK STATION WITH u > k GIVES A NEGATIVE F, whose logarithm is NaN and poisons the whole constant. The reference guards only against an INFINITE F (u == k exactly), not a negative one, so it propagates the NaN; that is reproduced rather than patched, because suppressing the term would quietly return a plausible number for a model the expansion does not cover. The condition cannot arise when every station has one server, but it can once s varies. Note that at Real precision Boost raises on log of a negative where double returns NaN, so the same input surfaces as an exception rather than a NaN; both are refusals and neither is a silent wrong number.

Arithmetic: TRANSCENDENTAL. Logs and factorials throughout.

Definition in file pfqn_xia.h.