6 service times occur according to the negative

6. Service times occur according to the negative exponential probability distribution.
7. The average service rate is greater than the arrival rate.
8. No more than one arrival can occur at a time.
3.1 Definition of birth-death process and its relation to queue
A birth-death process is a special type of Markov process. As the name implies, birth-
death processes were originally used to describe populations that were increased by birth
and decreased by death. Biologists, in particular, study the development of animals,
plants and other organisms by using birth-death processes. Later, it was found that birth-
death processes usually arise when there is a group of entities that are increased by birth
or arrivals and decreased by death or departures.
A queue, in the sense used here, can be described as a group of elements, such as
customers waiting for service, machines waiting for repair or cars waiting for at a petrol
station.

Introduction to Waiting Line Management MAH2021
In the birth-death processes, there is only one state variable, Q , namely, the number of
elements in the population or in the line. Both birth and death rates depend on Q . If Q = i ,
the birth rate is
i
 and the death rate is .
i
 Since, a birth increases the population by 1,
the rate at which Q increases from i to i +1 is
i
 ( the probability of increasing the
population). Similarly,
i
 (the probability of decreasing the population) is the rate at
which the population Q decreases from i to i – 1. The states 0, 1, 2, 3, … represent the
increase in population by 0, 1, 2, 3, … respectively. So, the only non-zero transition rates
are
i
 (when the population increases from i to i + 1) and
i
 ( when the population
decreases from i to i – 1). Transition matrix of the birth-death processes is shown below
. . . 3 2 1 0
.
.
.
3
2
1
0
























. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . 0 0 0
. . . 0 0 0
. . . 0 0 0
. . . 0 0 0 0
3 3
2 2
1 1
0
 
 
 

Rates always presume the existence of events, such as arrivals or at least changes of state.
Consequently, there can be no rate to go from state i to state i . There is, however, a rate
of leaving state i . This rate is equal to the sum of the rates of going from states i to i- 1 or i
+1, that is, .
i i
   In the literature, leaving rate of state i is defined by – ( ).
i i
   The
resulting transition matrix is then
. . . 3 2 1 0
A =
.
.
.
3
2
1
0
























 
 

. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
.. . . . 0 0 0
. . . 0 ) (
. 0 0 ) (
. . . 0 0 0
3 3
2 2 2 2
1 1 1 1
0 0
 
   
   
 
Note that the sum of the probabilities of changing from a state and changing to that state
is equal to zero. If ) , ,..., , , (
1 2 1 0 

i i
      be the probability that the system is in state
i , then for steady-state probability (probability of any state that remains the same), the
steady state equations will be
0  A 

Introduction to Waiting Line Management MAH2021
or 0
1 1 0 0
      
0 ) (
2 2 1 1 1 0 0
          
0 ) (
3 3 2 2 2 1 1
          
……………………………………
0 ) (
1 1 1 1 1
   
    i i i i i i
      
……………………………………
Addition of the 1
st
2 equations implies that
0
2 2 1 1
      
Addition of the 1
st
3 equations implies that
0
3 3 2 2
      
Similarly, sum of the 1
st
i