qsys_mmck.h File Reference

Exact analysis of the M/M/c/K queue (truncated Erlang form). More...

#include <cstddef>
#include <vector>
#include "line/api/qsys/qsys_types.h"
#include "line/num/number.h"
#include "line/util/error.h"

Include dependency graph for qsys_mmck.h:

Go to the source code of this file.

Classes
struct	line::qsys::MmckResult< T >

Namespaces
namespace	line
namespace	line::qsys

Functions
template<class T>
MmckResult< T >	line::qsys::qsys_mmck (const T &lambda, const T &mu, unsigned c, unsigned K)
	Exact analysis of the M/M/c/K queue (truncated Erlang form).

Detailed Description

Exact analysis of the M/M/c/K queue (truncated Erlang form).

Templated port of matlab/src/api/qsys/qsys_mmck.m, cross-checked against jar/src/main/java/jline/api/qsys/Qsys_mmck.java.

a = lambda/mu, rho = a/c p_n = a^n/n! p_0 0 <= n <= c p_n = a^c/c! rho^(n-c) p_0 c <= n <= K p_0 = 1 / [ sum_{n<c} a^n/n! + (a^c/c!) sum_{n=0}^{K-c} rho^n ]

All exponents are integers and the normalization is a finite sum, so the whole computation stays in the field of the inputs and is exact for T = Rational. The exact instantiation is not academic here: MATLAB builds the unnormalized vector as a^n/n! and warns about overflow, which is the failure mode for large a and K; in rational arithmetic the intermediate magnitudes are irrelevant.

Unlike the unbounded M/M/c, no stability condition is needed: a finite capacity makes every load admissible, rho >= 1 included.

Definition in file qsys_mmck.h.