PhPh: A Python package for PH/PH/c queueing systems

PhPh is a Python package for evaluating PH/PH/c queueing systems.

Features

• Available on PyPI.
• Allows for evaluation of multiserver models containing any Phase-type (PH) inter-arrival and service-time distribution.
• Returns fundamental performance metrics such as the expected waiting time, expected occupancy, and the probability of waiting longer than t units of time.
• Users can choose to view the metrics observed by actual arriving customers or virtual Poisson arrivals.

Quick start guide

The following shows how to get quickly started with PhPh.

Installation

Download and install PhPh directly from PyPI.

pip install phph

Usage

Start by specifying the inter-arrival and service-time distributions.

#Import packages
import phph
import numpy as np

#Set the server capacity
servers = 5

#ARRIVAL PARAMETERS - Example of Erlang distribution

#Initial distribution
arrivalInitDistribution = np.matrix([[1,0]])

#Phase-type generator
arrivalGenerator = np.matrix([[-12,12],
                              [0,-12]])

#SERVICE PARAMETERS - Example of hyper-exponential distribution

#Initial distribution
serviceInitDistribution = np.matrix([[(1/3),(1/2),(1/6)]])

#Phase-type generator
serviceGenerator = np.matrix([[-2,0,0],
                              [0,-1,0],
                              [0,0,-6]])

Now, create the model object.

mdl = phph.model(arrivalInitDistribution,arrivalGenerator,
                 serviceInitDistribution,serviceGenerator,
                 servers)

We can now use the object mdl to return various performance metrics. In the following, we calculate the expected waiting time, the expected occupancy, and the probability of waiting.

#Expected waiting time 
print(mdl.meanWaitingTime())
#0.455024

#Expected occupancy
print(mdl.meanOccupancy())
#6.896811

#Probability of waiting
print(mdl.probWait())
#0.562802

Classes and methods

The following shows how to create the model and provides a list of all available performance metrics.

Model class

Create the model object with:

mdl = phph.model(arrivalInitDistribution,arrivalGenerator,
                 serviceInitDistribution,serviceGenerator,
                 servers)

• arrivalInitDistribution is a NumPy row vector defining the initial distribution of the PH distribution associated with arrivals.
• arrivalGenerator is a NumPy matrix defining the PH generator of the distribution associated with arrivals.
• serviceInitDistribution is a NumPy row vector defining the initial distribution of the PH distribution associated with services.
• serviceGenerator is a NumPy matrix defining the PH generator of the distribution associated with services.
• servers is a non-zero positive integer and defines the number of servers in the queueing system.

Performance metrics

• mdl.meanWaitingTime(). Returns the actual (i.e. observed by arriving customers) expected waiting time.
• mdl.meanResponse(). Returns the actual expected total time in the system.
• mdl.probWait(type="actual"). Returns the probability of waiting. Choose between "actual" (default) and "virtual" using the argument type.
• mdl.probEmpty(type="actual"). Returns the probability that the system is empty. Choose between "actual" (default) and "virtual" using the argument type.
• mdl.probK(k,type="actual"). Returns the probability of observing k customers in the system on arrival. Choose between "actual" (default) and "virtual" using the argument type.
• mdl.waitDist(t,type="actual"). Returns the probability of waiting more than t time units. Choose between "actual" (default) and "virtual" using the argument type.
• mdl.meanQueueLength(). Returns the expected length of the queue.
• mdl.meanOccupancy(). Returns the expected occupancy.

Other output

• mdl.localStateDist(k). Returns the local state distribution of level k.
• mdl.localState(k). Returns the definition of the local state space of level k.