# Estimating cross K-function and envelopes for the marginal processes of a queue 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. ```r 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 $N_x$. - `T`: 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`. - `r`: 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`. ## 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. ## Returns A list with elements: - **r**: 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 $K_{ij}(r)$. - **T**: Length of the observed period of the processes. ## See Also `NHK`, `DepNHPPqueue` ## Examples ```r #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) ```

DepqueueNHK function