Package jline.solvers.fluid.analyzers

Class KoPenderAnalyzer

java.lang.Object
jline.solvers.fluid.analyzers.KoPenderAnalyzer
All Implemented Interfaces:
FluidAnalyzer
public class KoPenderAnalyzer extends Object implements FluidAnalyzer
Fluid and diffusion limits of the (MAP_t/Ph_t/inf)^N network of Y. M. Ko and J. Pender, "Diffusion limits for the (MAP_t/Ph_t/inf)^N queueing network", Oper. Res. Lett. 45 (2017) 248-253.

The mean and the covariance of the limit are integrated jointly: 

   dq/dt     = F(t,q) = A f(t,q)
   dSigma/dt = J Sigma + Sigma J' + G,   J = A df/dq,  G = A diag(f) A'

with A the jump matrix whose column e is the jump vector of event e and f the event rate vector. G is exactly dH dH' of Theorem 3.3, each independent Poisson term contributing l_e l_e' f_e. Where f is affine in q -- infinite-server stations and the arrival phase process -- J does not depend on q and both equations close exactly, so for the (MAP_t/Ph_t/inf)^N case the mean and the covariance are exact rather than asymptotic. Finite-server stations are admitted through the usual fluid min(x,c) term, where the covariance degrades to a linear-noise approximation.

This analyzer does NOT reuse the closing ODE. That formulation routes a departure from the source to the destination station and returns mass through the STATIONARY arrival-instant vector pie, replacing the D1' operator by the rank-one map pie*(D1*e)', i.e. by the PH renewal process with representation (pie, D0). Its stationary arrival rate is exact but its autocorrelation is gone, and a non-renewal arrival stream is the entire point of a MAP.

State layout, station-major, arrival phases before service phases: one u-block per (EXT station, class) holding the arrival MAP phase occupancy, which sums to 1, and one x-block per (queueing station, class) holding the fluid count in each service phase.