DepqueueNHK: Estimating cross K-function and envelopes for the marginal processes of a queue

Description

This function estimates the cross K-function between two (homogenous or nonhomogeneous) point processes in time, \(N_x\) and \(N_y\). It is evaluated in a grid of distances \(r\) and plotted. An envelope built by simulation under the hypothesis that the processes are the marginal processes (the input and the output) of a queue is also plotted.

It calls the auxiliary functions NHKaux, NHKaux2, NHKaux3 and DepqueueKenv, not intended for users.

Usage

DepqueueNHK(posx, posy, lambda,  T=NULL, nserv='infty', ddist='exp',argd=1,r=NULL, 
	typeEst=1, nsim=1000, conf=0.95,tit=NULL, cores=1,fixed.seed=NULL,...)

Arguments

posx

Numeric vector. Occurrence times of the points in the first point processes \(N_x\).

posy

Numeric vector. Occurrence times of the points in the second point processes \(N_y\).

lambda

Numeric vector. Intensity values of the input process \(N_x\).

Optional. Positive integer. Length of the period where the point are going to be generated. Only used in homogeneous processes (if lambda is a constant).

nserv

Optional. Number of servers in the queue system; only two values are possible: 1 or "infty" if the number of servers is infinity.

ddist

Character string. Identification of the probability distribution of the serving time. It must be one of the names of the probability distributions available in the stats package, see Distributions; for example "gamma", "geom", etc.

argd

Optional. Numeric value or vector containing the arguments to be used in the probability distribution in ddist.

Optional. Numeric vector. Grid values where the K-function must be evaluated. If it is NULL, a default vector is used; see Details.

typeEst

Optional. Two possible values: 1 or 2. They determine which one of the two available estimators of the function \(K_{ij}\) has to be used; see Details.

nsim

Optional. Numeric value. Number of simulations to obtain the envelope.

conf

Optional. Numeric value in (0,1). Confidence level of the envelope for the K-function.

tit

Optional. Title to be used in the plot of the K-function.

cores

Optional. Number of cores of the computer to be used in the calculations.

fixed.seed

An integer or NULL. If it is an integer, that is the value used to set the seed in random generation processes. It it is NULL, a random seed is used.

...

Further arguments to be passed to the function plot.

Value

A list with elements:

Vector of values \(r\) where the cross K-function is estimated.

NHKr

Estimated values of \(K_{ij}(r)\).

KenvL

Lower bounds of the envelope of\(K_{ij}(r)\).

KenvU

Upper bounds of the envelope of \(K_{ij}(r)\).

Length of the observed period of the processes.

Details

This function estimates the cross K function between two (homogenous or nonhomogeneous) point processes in time, \(N_x\) and \(N_y\). Two different estimators are available, see NHK for details.

An envelope for the cross K function is built under the hypothesis that the processes are the input and the output processes of a queue. The envelope is based on simulations, where processes generated by DepNHPPqueue are used. The distribution of the serving time is specified in arguments ddist and argd. The queue can have one or infinity servers.

If argument r is NULL, the following r-grid is used to evaluate the function

r1<-max(20, floor(T/20))

r<-seq(1,r1,by=2)

if (length(r)>200) r<-seq(1,r1,length.out=200)

where T is the length of the observed period.

Examples

Run this code

# NOT RUN {
#Observed process: independent Poisson processes
set.seed(123)
T<-1000
lambda1<-runif(T,0,0.05)
dendist<- dexp(c(1:T), 1)
lambda2<-convolve(lambda1,rev(dendist),type='o')[1:T]
posx<-simNHPc(lambda = lambda1, fixed.seed = 134)$posNH
posy<-simNHPc(lambda = lambda2, fixed.seed = 135)$posNH

DepqueueNHK(posx=posx, posy=posy, lambda=lambda1,  ddist='exp',argd=1, 
	r=seq(1,40, by=5), fixed.seed=123,nsim=500)

#Observed process: input and output processes  of a queue
#aux<-DepNHPPqueue(lambda=lambda1, d=2, fixed.seed=123, argd=1, dplot=FALSE)$posNH
#DepqueueNHK(posx=aux[[1]], posy=aux[[2]], lambda=lambda1,  ddist='exp',argd=1, 
#	r=seq(1,40, by=5), fixed.seed=123)

	

# }