A wide range of
situations, such as warehousing and cross-docking operations,
population infestations, and the servicing of customers on
a hotline, can be described in terms of fairly simple stochastic
processes. This section concentrates on an important
special case called the birth-death process.
For the birth-death process, the population is the number
of entities that comprise some system. The population ranges
from 0 to a specified maximum population. Population changes
with either a birth event or a death event. A birth
increases the population by one, and a death decreases the
population by one. The figure below shows the state network
for a birth death system. The numbers in the circles indicate
the population state variable.
State Network when the system is purged when
an arrival occurs at the maximum population
The times between events are governed by exponential distributions.
The birth rate is the parameter of the exponential distribution
that describes the time to the next birth. The birth rate is
the inverse of the mean time between birth events. The death
rate is the parameter of the exponential distribution that
governs the time to the next death. The death rate is
the inverse of the mean time between death events. Birth and
death rates may be constant or depend on the population value.
Since the maximum population is finite, it is necessary to
define what happens when the population reaches this maximum.
Various alternatives could be modeled. The process
could simply terminate. The process could continue to operate,
but births would be neglected. Our models have two options, the
*purge* option and the *balk* option. The purge
option assumes that when the maximum population is reached, the
system is *purged* with the next arrival.
That is, the population returns to state 0. This is called the
*purge* event. There is a fixed cost of the purge as well
as a variable cost that is linear with the number purged. The purge
model could represent a batch process that is triggered when the
population reaches a maximum amount.
The *balk* option assumes that when the process reaches
the maximum, the next arrival will balk and not enter the system.
There is a charge for this event to represent the cost of the
lost arrival. If some other rule of operation is more appropriate,
the model can be rather easily changed to represent the alternative.
State Network when arrivals balk when the population
is at the maximum value
Although we use the terms population, birth, and death, the
models described here not limited to biological populations.
Many other applications are readily apparent. For example we
could consider a call center where the population is the number
of customers waiting or in service, a birth is the arrival of
a new call, and a death is the completion of a service.
Several different models are described on this page. The CTMC
and the DTMC involve no decisions. The purpose of an analysis
is to determine the probability distribution of the states. The
MDP adds the action of either purge or balk to each state and
dynamic programming determines the optimum decision at each state. |