IndTestPP (version 3.0)

DepNHPPqueue: Generating dependent point processes by a tandem queueing network

Description

It generates \(d\) dependent (homogeneous or nonhomogeneous) Poisson processes using \(d-1\) queues in tandem.

Usage

DepNHPPqueue(lambda, d,   T=NULL,  nEv=NULL,  nserv="infty", Clambda=TRUE,
	            ddist='exp',argd=1,dplot=TRUE, fixed.seed=NULL, ...)

Arguments

lambda

Numeric value or vector. Intensity of the first Poisson process. If its length is 1, homogeneous processes are generated.

d

Integer. Number of dependent processes to be generated.

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).

nEv

Optional. Positive integer. Number of points to be generated in the processs. Only used in homogeneous processes.

Clambda

Optional logical flag. Only used in nonhomogeneous processes. If it is TRUE, the intensity vectors of the generated point processes is calculated.

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. A value, vector or a matrix (with d-1 rows) containing the arguments to be used in the probability distribution in ddist; see Details.

dplot

Optional. A logical flag. If it is TRUE, the marginal processes are plotted.

fixed.seed

Optional. An integer or NULL. Value used to set the seed in random generation processes; if it is NULL, a random seed is used.

...

Further arguments to be passed to the function plot.

Value

A list with elements

posNH

A list of \(d\) vectors, containing the occurrence points in each point process. The name of the elements in the list are N1, N2,..., Nd.

lambdaM

A \(d\)-column matrix containing the intensity vectors of the \(d\) dependent processes, in nonhomogeneous processes and if Clambda=TRUE. Otherwise, the input argument lambda.

Details

The generation algorithm is described in Cebrian et al. (2020). The generation of dependent homogeneous Poisson processes can be based on a M\ M\ 1 or a M\ M \ \(\infty\) queue, and the intensity of each output process is equal to the intensity of the input process, see Burke(1956). The generation of dependent nonhomogeneous Poisson processes is based on a M(t)\ G \ \(\infty\) queue and the intensity of each output process is equal to the convolution \(\lambda_{out}(t)=\lambda_{inp}(t)*f(t)\), where \(\lambda_{inp}(t)\) is the input intensity and \(f(t)\) the density function of the service time, see Keilson and Servi (1994).

In the homogeneous processes, the argument \(\lambda\) can be an integer or a vector with equal values. In the first case, the argument nEv or T must be specified; in the second, the length of the vector determines the length of period where the points are generated (as in the nonhomogeneous case).

The serving times in all the queues must have a probability distribution in the same family (ddist) but the parameters of that distribution can be different in each queue. If the parameters are the same in all the queues, argument argd is a numeric value (if there is only one parameter) or a vector; if the parameters are different in each queue, argd must be a matrix with (d-1) rows, even if there is only one parameter it must be a (d-1) \(\times\) 1 matrix.

The occurrence times in each output process are sorted, so that the output time in row i does not always correspond to the input time in the same row. The number of points in the output process may be lower than in the input (if some outputs have not been observed).

References

Burke, P. J. (1956). The Output of a Queuing System. Operations Research. 4(6), 699-704.

Cebrian, A.C., Abaurrea, J. and Asin, J. (2020). Testing independence between two point processes in time. Journal of Simulation and Computational Statistics

Keilson, J. Servi, L.D. (1994). Networks of nonhomogeneous M G \(\infty\). J. Appl. Probab., 31, 157-68.

See Also

IndNHPP, DepNHNeyScot, DepNHPPMarked, DepNHCPSP

Examples

# NOT RUN {
#Generation of 3 dependent HPPs, with  one server  and exponential service time equal to 10
aux<-DepNHPPqueue(lambda=0.05, d=3,  nEv=25,fixed.seed=123,nserv=1, argd=0.1)
aux$posNH

#Generation of 4 dependent NHPPs, with infinity servers and different mean service times
#at each queue
lambda<-runif(200,0,0.1)
aux<-DepNHPPqueue(lambda=lambda, d=4, fixed.seed=123, argd=cbind(c(0.1, 0.3, 0.1)))
aux$posNH


#Generation of 3 dependent NHPPs, with infinity servers and  Gamma service times 
#with different parameters at each queue
lambda<-runif(200,0,0.1)
aux<-DepNHPPqueue(lambda=lambda, d=3, ddist='gamma',fixed.seed=123, 
                  argd=cbind(c(0.1, 0.3), c(1,2)))
aux$posNH

# }